矩阵带宽缩减技术在隐式间断有限元中的应用

doi:10.13700/j.bh.1001-5965.2019.0281

北京航空航天大学学报 ›› 2020, Vol. 46 ›› Issue (3): 532-540.doi: 10.13700/j.bh.1001-5965.2019.0281

矩阵带宽缩减技术在隐式间断有限元中的应用

李亮, 吴颂平

北京航空航天大学航空科学与工程学院, 北京 100083

收稿日期:2019-06-05 发布日期:2020-03-28
通讯作者: 吴颂平 E-mail:wusping825@163.com
作者简介:李亮,男,博士研究生。主要研究方向:间断有限元格式在计算流体力学中的应用;吴颂平,男,博士,教授,博士生导师。主要研究方向:计算流体力学。
基金资助:
国家自然科学基金（91530325）

Application of matrix bandwidth reduction technique in implicit discontinuous Galerkin

LI Liang, WU Songping

School of Aeronautic Science and Engineering, Beihang University, Beijing 100083, China

Received:2019-06-05 Published:2020-03-28
Supported by:
National Natural Science Foundation of China (91530325)

摘要/Abstract

摘要： 为了数值求解二维Euler方程，以间断有限元方法作为空间离散、向后差分公式（BDF）作为时间离散。针对采用牛顿法求解源于隐式时间积分的非线性方程组，构造了相应的Jacobi矩阵，其具有阶数高、稀疏性强、数值非对称的特点。在每个时间步内，选择带预处理的广义极小残量（GMRES）方法求解线性方程组，预处理矩阵由不完全LU分解（ILU）方法构造。将矩阵带宽缩减技术应用于上述求解过程，无需额外的存储空间，就缩小了预处理矩阵与系数矩阵的差距，从而加快了GMRES方法的收敛、增大了可用的时间步长。通过求解典型的空气动力学问题，检验了该应用的有效性。

关键词: 间断有限元, 隐式方法, 线性方程组, 广义极小残量(GMRES)方法, 矩阵带宽缩减

Abstract: To numerically solve the two-dimensional Euler equations, discontinuous Galerkin method and backward difference formula (BDF) are used as spatial and temporal discretization, respectively. The Newton-Raphson method is taken to solve the nonlinear equations arising from the implicit time integration. The Jacobia matrix is constructed. Owing to the high-order, sparsity and non-symmetry of the matrix, the preconditioned generalized minimal residual (GMRES) method is chosen in every time step for solving the linear equations. The preconditioner is constructed using incomplete lower-upper (ILU) decomposition method. The bandwidth reduction technique is applied to the solution of the linear equations. Without extra storage cost, the application narrows the difference between the preconditioner and the coefficient matrix, thus accelerating the convergence of GMRES method and increasing the available time step size for temporal integration. Typical aerodynamic problems are solved to test the effectiveness of the application.

Key words: discontinuous Galerkin, implicit scheme, linear equations, generalized minimal residual (GMRES) method, matrix bandwidth reduction

中图分类号:

O242.1
O35

李亮, 吴颂平. 矩阵带宽缩减技术在隐式间断有限元中的应用[J]. 北京航空航天大学学报, 2020, 46(3): 532-540.

LI Liang, WU Songping. Application of matrix bandwidth reduction technique in implicit discontinuous Galerkin[J]. Journal of Beijing University of Aeronautics and Astronautics, 2020, 46(3): 532-540.

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矩阵带宽缩减技术在隐式间断有限元中的应用

Application of matrix bandwidth reduction technique in implicit discontinuous Galerkin

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